A. The Fundamental Difference
| Property |
Rational Numbers |
Irrational Numbers |
| Fraction Form |
Can be written as p/q |
CANNOT be written as p/q |
| Decimal Form |
Terminating or Repeating |
Non-terminating & Non-repeating |
| Examples |
1/2, 0.5, 0.333..., √9 |
√2, √3, π, 0.10110111... |
B. Identifying Irrational Roots (Surds)
If n is a natural number but NOT a perfect square (1, 4, 9, 16...), then √n is always irrational.
- √2 ≈ 1.4142... (Irrational)
- √5 ≈ 2.2360... (Irrational)
- √25 = 5 (Rational)
1. Define an irrational number.
A number that cannot be expressed as a ratio of two integers and has a non-terminating, non-repeating decimal form.
2. Is √2 rational or irrational?
Irrational. 2 is not a perfect square.
3. Is √49 irrational?
No, it is Rational. √49 = 7.
4. Classify π.
Irrational.
5. Is 0.121212... (repeating) irrational?
No, it is Rational because it has a repeating pattern.
6. Between which two integers is √2 located?
1 and 2 (Since √1 < √2 < √4).
7. True or False: All decimals are rational.
False. Non-terminating, non-repeating decimals are irrational.
8. Is √0.25 rational?
Yes. √0.25 = 0.5 (which is 1/2).
9. What is the value of √2 to three decimal places?
1.414 (approximately).
10. Can an irrational number be written as p/q?
No. That is the definition of a rational number.
11. Is √8 irrational?
Yes. 8 is not a perfect square.
12. Identify the rational number: √3, √5, √9, √11.
√9 (which is 3).
13. Does the decimal expansion of π ever end?
No. It is non-terminating.
14. True or False: The sum of two irrational numbers is always irrational.
False. For example, √2 + (-√2) = 0, which is rational.
15. Is √20 irrational?
Yes.
16. What is √1? Rational or Irrational?
Rational (√1 = 1).
17. Identify: 0.101001000...
Irrational. It has a pattern but it is not repeating the same sequence.
18. Is √100 rational?
Yes (10).
19. Can irrational numbers be found on the number line?
Yes. They represent specific points just like rational numbers.
20. Is the product of √2 and √2 rational or irrational?
Rational. √2 × √2 = 2.